Unfortunately, it's not common to be able to fit a 19" PC monitor into a 19" CGA monitor frame. Never hurts to try, though. Still, I wouldn't count on it.
Depending on the cabinet, using a de-cased monitor can either be a mild pain in the butt or a major frustration. (Of course, some people actually welcome those challenges.)
The easiest cabinet to mount in would be one that uses a horizontal-mount monitor in a horizontal orientation. These provide you with a flat place to set your chassis (since if you de-case the monitor, you'll have a seperate tube and chassis).
The second-easiest cabinet would be a universal-mount, horizontal oriented cabinet because you won't have a sehlf, but you'll be able to build one with little effort.
From personal experience, mounting a 27" de-cased monitor in a horizontal position has actually proved easier then mounting a 19" de-cased monitor in a vertical position.
With any of those methods I've mentioned, it requires completed removing a monitor from its case. If you wanted to avoid that, you might be able to get away with mounting a monitor (case and all) directly into a machine using a shelf if you were to remove the pedestal. I've never tried that route. Of course, you might be able to tell from the outside.